Quadratic Reciprocity in Odd Characteristic
نویسنده
چکیده
The answer to questions like this can be found with the quadratic reciprocity law in F[T ]. It has a strong resemblance to the quadratic reciprocity law in Z. We restrict to F with odd characteristic because when F has characteristic 2 every element of F[T ]/(π) is a square, so our basic question is silly in characteritic 2. (There is a good analogue of quadratic reciprocity in characteristic 2, but we don’t discuss it here.) In Section 2, we define the Legendre symbol in F[T ], establish some of its properties, and state the quadratic reciprocity law. The proof of the law is in Section 3. Some applications are given in Section 4 and a little history behind the reciprocity law is in Section 5. Throughout our discussion, F will denote a finite field with odd characteristic and size q. For a nonzero polynomial f ∈ F[T ], we set Nf = #F[T ]/(f) = q f ,
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